Existence of global solutions and energy decay for the Carrier equation with dissipative term

Abstract

We prove the existence and uniqueness of global solutions to the mixed problem for the Carrier equation $$ u_{tt} - M(\int_{\Omega} u^{2}\, dx ) \Delta u + g(u_{t}) = f, $$ where $g^{\prime}(s) \geq 0, \, 0 < m_{0} \leq M(\lambda)$ and no "smallness" conditions are imposed on the initial data. Moreover, the algebraic and exponential decays of the energy are proved.